A First Course in Differential Equations with Modeling Applications 11th edition Dennis G. Zill - Solutions

A critical point c of an autonomous first-order DE is said to be isolated if there exists some open interval that contains c but no other critical point. Can there exist an autonomous DE of the form given in (2) for which every critical point is nonisolated? Discuss; do not think profound
Consider the autonomous DE dy/dx = (2/πy)y - sin y. Determine the critical points of the equation. Discuss a way of obtaining a phase portrait of the equation. Classify the critical points as asymptotically stable, unstable, or semi-stable.
Consider the autonomous differential equation dy/dx = f (y), where the graph of f is given. Use the graph to locate the critical points of each differential equation. Sketch a phase portrait of each differential equation. By hand, sketch typical solution curves in the subregions in the xy-plane
Find the critical points and phase portrait of the given autonomous first-order differential equation. Classify each critical point as asymptotically stable, unstable, or semi-stable. By hand, sketch typical solution curves in the regions in the xy-plane determined by the graphs of the
Consider the autonomous rst-order differential equation dy/dx = y2 - y4 and the initial condition y(0) = y0. By hand, sketch the graph of a typical solution y(x) when y0 has the given values.(a) y0 > 1(b) 0 < y0 < 1(c) -1 < y0 < 0(d) y0 < -1
Consider the autonomous first-order differential equation dy/dx = y - y3 and the initial condition y(0) = y0. By hand, sketch the graph of a typical solution y(x) when y0 has the given values.(a) y0 > 1(b) 0 < y0 < 1(c) -1 < y0 < 0(d) y0 < -1
(a) Identify the nullclines (see Problem 17) in Problems€‘1, 3, and 4. With a colored pencil, circle any lineal elements in figures (1), (2) and (3) that you think may be a lineal element at a point on a nullcline.(b) What are the nullclines of an autonomous first-order
For a first-order DE dy/dx = f (x, y) a curve in the plane defined by f (x, y) = 0 is called a nullcline of the equation, since a lineal element at a point on the curve has zero slope. Use computer software to obtain a direction field over a rectangular grid of points for dy/dx = x2 - 2y, and
(a) Consider the direction field of the differential equation dy/dx = x(y - 4)2 - 2, but do not use technology to obtain it. Describe the slopes of the lineal elements on the lines x = 0, y = 3, y = 4, and y = 5.(b) Consider the IVP dy/dx = x(y - 4)2 - 2, y(0) = y0, where y0 < 4. Can a
In parts (a) and (b) sketch isoclines f (x, y) = c for the given differential equation using the indicated values of c. Construct a direction field over a grid by carefully drawing lineal elements with the appropriate slope at chosen points on each isocline. In each case, use this rough direction
The given figure represents the graph of f (y) and f (x), respectively. By hand, sketch a direction field over an appropriate grid for dy/dx = f (y) (Problem 13) and then for dy/dx = f (x) (Problem 14).1. 2. ft y х
Use computer software to obtain a direction field for the given differential equation. By hand, sketch an approximate solution curve passing through each of the given points.1. y' = x(a) y(0) = 0(b) (-2)2. y' = x + y(a) y(-2) = 2(b) y(1) -33. y dy/dx = -x(a) y(1) = 1(b) y(0) = 44. dy/dx = 1/y(a)
Reproduce the given computer-generated direction field. Then sketch, by hand, an approximate solution curve that passes through each of the indicated points. Use different colored pencils for each solution curve.1. dy/dx = x2 €“y2(a) y(-2) = 1(b) y(3) = 0(c) y(0) = 2(d) y(0) =

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